Common Mistakes
Traps in Binomial Theorem
5 mistake patterns students fall for. Each one shows the wrong approach vs the correct approach.
Off-by-one in general term
FORMULAThe general term formula gives T_{r+1}, not T_r. The 5th term uses r=4.
✗ WRONG: 5th term of (a+b)^n: T_5 = nC5 · a^{n-5} · b^5
✓ RIGHT: 5th term: T_5 = T_{4+1} = nC4 · a^{n-4} · b^4. Always use r = (term number - 1).
Missing negative signs in expansion
SIGN ERRORWhen expanding (a-b)^n, the sign alternates. T_{r+1} has factor (-1)^r.
✗ WRONG: T_3 of (x-2)^5 = 5C2 · x^3 · 4 = 40x^3
✓ RIGHT: T_3 = 5C2 · x^3 · (-2)^2 = 40x^3. Here it's positive, but for odd r it's negative. Track (-1)^r carefully.
Integral terms: forgetting both exponents must be integers
CASE MISSFor a term to be rational, BOTH exponents in the general term must be non-negative integers.
✗ WRONG: In (2^{1/3} + 3^{1/4})^12, checking only that r is a multiple of 4
✓ RIGHT: Need r to be a multiple of 4 AND (12-r) to be a multiple of 3. Check both conditions simultaneously.
Sum of coefficients vs binomial coefficients
FORMULASum of binomial coefficients of (1+x)^n is 2^n. Sum of coefficients of f(x) is f(1).
✗ WRONG: Sum of coefficients of (1-3x+10x^2)^n = 2^n
✓ RIGHT: Put x=1: (1-3+10)^n = 8^n. The sum of coefficients equals f(1), not always 2^n.
Remainder: wrong base decomposition
SIGN ERRORWhen finding remainder of a^n mod m, the decomposition must use the correct sign.
✗ WRONG: 64^k mod 9: writing 64 = 9×7 + 1, then remainder = 1^k (correct here, but sign errors common)
✓ RIGHT: Always verify: base = divisor × quotient + remainder. 64 = 9×7 + 1 ✓. Then expand (9×7+1)^k.